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Comparing two sample means

Comparing two sample means. Dr David Field. Comparing two samples. Researchers often begin with a hypothesis that two sample means will be different from each other In practice, two sample means will almost always be slightly different from each other

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Comparing two sample means

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  1. Comparing two sample means Dr David Field

  2. Comparing two samples • Researchers often begin with a hypothesis that two sample means will be different from each other • In practice, two sample means will almost always be slightly different from each other • Therefore, statistics are used to decide whether the observed difference between two samples is meaningful or not • To do this, we test the null hypothesis that the two samples were both drawn randomly from the same population

  3. Test statistics • To test the null hypothesis we need to quantify the strength of the evidence against it • This is done using test statistics • when the test statistic is larger, there is more evidence against the null hypothesis • What makes test statistics different from other statistics is that they have known probability distributions when the null hypothesis is true • we know the p of a test statistic of 1 or >1 occurring purely due to sampling variation from a null distribution • the p of a test statistic of 2 or > 2 will be lower than the p of a test statistic of >1 • if the p of the test statistic occurring purely due to sampling variation is < 0.05 (5%) the null hypothesis is rejected • Test statistics with known probability distributions under the null hypothesis include z, t, r, and chi-square • Mean, Median, SD are not test statistics

  4. Confidence intervals as a test • Lecture 2 explained how to calculate a 95% confidence interval around a single sample mean • this was achieved using the SE of an inferred sampling distribution of the mean • collecting two samples and calculating two separate confidence intervals establishes that the two samples are from different populations if the confidence intervals do not overlap • but it does not allow a conclusion to be reached when the confidence intervals do overlap • To calculate a test statistic to directly test the null hypothesis we need to consider a slightly different sampling distribution • the sampling distribution of the difference between two means

  5. Sampling distribution of the difference between two means • Normally, you are only able to measure 2 samples and calculate 2 means and the difference between them • But test statistics are based on properties of an assumed underlying sampling distribution of the difference between two means • The best way to understand test statistics is to consider unusual or artificial examples where full population data and sampling distributions are available • Therefore….

  6. The two populations Mean 4.0 SD 0.4 Mean 4.5 SD 0.8

  7. Sampling distribution of the difference between two means 9.4.1 • Take a large number of samples of 5 cats from the UK population • Arrange the samples in pairs and for each pair calculate the difference between the two means • Half the differences will be negative and half of them will be positive • Therefore the mean of this sampling distribution will be zero. This differs from the sampling distribution of a single sample mean, which has a mean equal to the underlying population mean • The sampling distribution of the difference between two means will be normally distributed

  8. ORIGINAL DISTRIBUTION is the population frequency distribution of weight differences between pairs of individual cats • Black solid curves are sampling distributions of weight differences between 2 sample means, for samples of of 4, 16, and 64 cats 1 SE 1 SD of population

  9. Standard error of the difference between two sample means • σ (sigma) means the SD of the population of difference scores • N1 and N2 are the two sample sizes • the formula allows the SE of the sampling distribution to be calculated when the two samples differ in size • Like the SE of a single sample mean, this SE gets smaller as N increases and gets smaller as the SD gets smaller • Smaller SE makes it easier to reject null hypothesis σ 1 1 = SE + N1 N2

  10. SE of the difference between mean Kg for two samples of 5 UK cats • 1/5 (or 1/2, or 1/3, or 1/20) is a number less than 1 • The square root makes the number larger, but never makes it greater than 1 • So, the population SD gets multiplied by a number smaller than 1, which is why the SE is always smaller than the SD of the population 1 1 = 0.8 0.506 Kg + 5 5

  11. For the highlighted pair of samples the difference between the means is 0.5Kg • What percentage of sample pairs have a difference of 0.5Kg or larger? • If we expressed the difference of 0.5Kg in units of SE we could answer that question • This is because the converted score is a Z score Remember that in this theoretical example we know that both samples are from the same population, and the purpose is to calculate the p of a difference this big or bigger occurring when that is the case

  12. Converting the difference between 2 sample means to a Z score The difference between the means 0.5 Z = 1 0.8 1 + The SE formula 5 5 Z = 0.99

  13. 16.1% of the total area under the normal curve corresponds to values of 0.99 or greater 16.1% of differences between means of sample size 5 will have Z scores greater than 0.99

  14. From Z back to Kg • So, 16.1% of differences between pairs of samples of N=5 drawn from the population of UK cats will be 0.5Kg or larger • This is the same as saying the probability of a single comparison producing a difference of 0.5Kg or greater is 16.1%

  15. What if the population SD (σ) is unknown? • Usually, researchers only have two samples to compare, and the population parameters are unknown. • In this situation the sample SD is used instead of the population SD, and the SE formula is modified SD12 SD22 SE = + N1 N2

  16. For the highlighted pair of samples the mean difference is 0.5Kg • The sample SD’s will be used in the modified formula instead of the unknown population SD

  17. Converting the difference between 2 means to a Z score when σ is unknown 0.5 Z = 0.52 0.72 + 5 5 0.5 = 1.29 0.38

  18. How much evidence is there against the null hypothesis? • 9.8% of Z statistics are > 1.29, so we would not conclude that the two samples of cats are from different countries if we used the 5% cut off • In this example, we know that the two samples were from the same population, so we can verify that this was the correct conclusion • On the other hand, if two samples had a mean difference of 0.8Kg, then assuming the sample SD’s remain the same, the resulting Z statistic would be 2.07 • Only 1.9% of Z statistics are greater than 2.07, and if we didn’t know that the two samples came from the same population we would reject the null hypothesis, and by doing so commit a Type I error

  19. The Z score of the difference between samples of 5 UK and 5 Greek cats 4.1 – 3.7 Z = 0.52 0.32 + 5 5 0.4 = 1.53 0.26

  20. How much evidence is there against the null hypothesis? • 6.3% of Z statistics are > 1.53, so we would be unable to conclude that the two samples of cats are from different countries if we used the 5% cut off • In this example we know that the two samples were from different populations, so we have committed a Type II error by failing to reject the null hypothesis • Type II errors like this are common when the sample size is small

  21. The Z score of the difference between samples of 12 UK and 12 Greek cats 4.6 – 4.1 Z = 0.62 0.22 + 12 12 0.5 = 2.73 0.18

  22. How much evidence is there against the null hypothesis? • 0.032% of Z statistics are > 2.73, so we would conclude that the two samples of cats are from different countries if we used the 5% cut off • In this example we know that the two samples were from different populations, so we have correctly rejected the null hypothesis

  23. Important caveat • What I have described today is called a “Z test” • But, the formula for estimating the SE of the difference between 2 means used in the Z test is only accurate when the individual sample sizes are 30 or more • This is because the estimate of the population SD is not accurate • There is a different test that uses an accurate estimate of the SE when sample size is less than 30 • the “t test”, which is covered in the next lecture • Because the t test produces the same results as the Z test when the sample size is >30 computer programs like SPSS generally only give the option of a t test • Both tests work on the same principle, but the Z test is less complicated and easier to understand

  24. General principle of test statistics 2.6.1 • All test statistics have known probability distributions when variation in the DV due to the IV is zero (i.e. the null hyp is true) • Z has the distribution of the standard normal distribution • Other test statistics have different shaped distributions, and different calculation formulas, but the general principle for converting the test statistic to a p value is the same. variation in the DV due to the IV = test statistic other variation in the data (error)

  25. List of statistical terms for revision • This lecture made use of terms introduced in previous lectures, and only introduced one new term • sampling distribution of the difference between two means

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